We are open Saturday and Sunday!

Call Now to Set Up Tutoring:

(888) 888-0446

Precalculus : Rational Functions

Study concepts, example questions & explanations for Precalculus

Create An Account

All Precalculus Resources

12 Diagnostic Tests 380 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #1 : Find A Point Of Discontinuity

What are the holes or vertical asymptotes, if any, for the function: $y=\frac{x^2-1}{x-1}$

Possible Answers:

$\textup{Hole at }x=1,-1$

$\textup{There are no discontinuities.}$

$\textup{Hole at } x=1; \textup{ No vertical asymptotes.}$

$\textup{Hole at }x=1; \textup{Vertical asymptote at } x=-1$

$\textup{Hole at } x=-1; \textup{ No vertical asymptotes.}$

Correct answer:

$\textup{Hole at } x=1; \textup{ No vertical asymptotes.}$

Explanation:

Factorize the numerator for the function:

$y=\frac{x^2-1}{x-1}$

$y=\frac{(x+1)(x-1)}{x-1} = x+1$

The removable discontinuity is x-1 since this is a term that can be eliminated from the function. There are no vertical asymptotes.

Set the removable discontinutity to zero and solve for the location of the hole.

x-1=0

x=1

The hole is located at: x=1

Report an Error

Example Question #1 : Find A Point Of Discontinuity

For the following function, $y=\frac{x^2-6x+9}{x^2-9}$ , find all discontinuities, if possible.

Possible Answers:

$\textup{Asymptote at: } x=3 \textup{; Hole at: x=-3}$

$\textup{Asymptote at: }x=\pm3$

$\textup{Asymptote at: } x=-3 \textup{; Hole at: x=3}$

$\textup{Hole at: }x=\pm3$

$\textup{There are no discontinuities.}$

Correct answer:

$\textup{Asymptote at: } x=-3 \textup{; Hole at: x=3}$

Explanation:

Rewrite the function $y=\frac{x^2-6x+9}{x^2-9}$ in its factored form.

$y=\frac{x^2-6x+9}{x^2-9}=\frac{(x-3)(x-3)}{(x+3)(x-3)}$

Since the x-3 term can be cancelled, there is a removable discontinuity, or a hole, at x=3 .

The remaining denominator of x+3 indicates a vertical asymptote at x=-3 .

Report an Error

Example Question #1 : Find A Point Of Discontinuity

If possible, find the type of discontinuity, if any:

$y= \frac{6x-9}{2x-3}$

Possible Answers:

$\textup{There are no discontinuities.}$

$\textup{Hole at }x=\frac{3}{2}$

$\textup{Hole and asymptote at }x=\frac{3}{2}$

$\textup{Asymptote at }x=\frac{3}{2}$

$\textup{Hole at }x=-\frac{3}{2}$

Correct answer:

$\textup{Hole at }x=\frac{3}{2}$

Explanation:

By looking at the denominator of $y= \frac{6x-9}{2x-3}$ , there will be a discontinuity.

Since the denominator cannot be zero, set the denominator not equal to zero and solve the value of .

$2x-3\neq 0$

$2x\neq 3$

$x\neq\frac{3}{2}$

There is a discontinuity at $x=\frac{3}{2}$ .

To determine what type of discontinuity, check if there is a common factor in the numerator and denominator of $y= \frac{6x-9}{2x-3}$ .

Since the common factor is existent, reduce the function.

$y= \frac{6x-9}{2x-3}=\frac{(3)(2x-3)}{2x-3}=3$

Since the 2x-3 term can be cancelled, there is a removable discontinuity, or a hole, at $x=\frac{3}{2}$ .

Report an Error

Example Question #2 : Find A Point Of Discontinuity

Find the point of discontinuity for the following function:

$m(x)=\frac{x^2+4x-5}{x^2+7x+10}$

Possible Answers:

There is no point of discontinuity for the function.

(0, 4)

(-5, 2)

(5, 1)

Correct answer:

(-5, 2)

Explanation:

Start by factoring the numerator and denominator of the function.

$m(x)=\frac{x^2+4x-5}{x^2+7x+10}=\frac{(x+5)(x-1)}{(x+5)(x+2)}=\frac{x-1}{x+2}$

A point of discontinuity occurs when a number is both a zero of the numerator and denominator.

Since x=-5 is a zero for both the numerator and denominator, there is a point of discontinuity there. To find the value, plug in into the final simplified equation.

$\frac{-5-1}{-5+2}=2$

(-5, 2) is the point of discontinuity.

Report an Error

Example Question #1 : Find A Point Of Discontinuity

Find the point of discontinuity for the following function:

$p(x)=\frac{x^3-4x^2-5x}{x^2-9x+20}$

Possible Answers:

There is no point fo discontinuity for this function.

(-1, 0)

(2, -3)

(5, 30)

Correct answer:

(5, 30)

Explanation:

Start by factoring the numerator and denominator of the function.

$p(x)=\frac{x^3-4x^2-5x}{x^2-9x+20}=\frac{x(x^2-4x-5)}{(x-5)(x-4)}=\frac{x(x-5)(x+1)}{(x-5)(x-4)}=\frac{x(x+1)}{x-4}$

A point of discontinuity occurs when a number is both a zero of the numerator and denominator.

Since x=5 is a zero for both the numerator and denominator, there is a point of discontinuity there. To find the value, plug in into the final simplified equation.

$\frac{5(5+1)}{5-4}=30$

(5, 30) is the point of discontinuity.

Report an Error

Example Question #6 : Find A Point Of Discontinuity

Find a point of discontinuity for the following function:

$r(x)=\frac{16-4x^2}{x^2-4}$

Possible Answers:

There are no discontinuities for this function.

(2, 4)

(-2, -4)

(-4, -2)

Correct answer:

(-2, -4)

Explanation:

Start by factoring the numerator and denominator of the function.

$r(x)=\frac{16-4x^2}{x^2-4}=\frac{-4(x^2-4)}{(x+2)(x-2)}=\frac{-4(x-2)(x+2)}{(x-2)(x+2)}=-4$

A point of discontinuity occurs when a number is both a zero of the numerator and denominator.

Since $x=\pm2$ is a zero for both the numerator and denominator, there is a point of discontinuity there. Since the final function is r(x)=-4 , (2, -4) and (-2, -4) are points of discontinuity.

Report an Error

Example Question #7 : Find A Point Of Discontinuity

Find a point of discontinuity for the following function:

$s(x)=\frac{180-5x^2}{x^2-36}$

Possible Answers:

(-6, -5)

(-5, -6)

There are no points of discontinuity for this function.

(6, 5)

Correct answer:

(-6, -5)

Explanation:

Start by factoring the numerator and denominator of the function.

$s(x)=\frac{180-5x^2}{x^2-36}=\frac{-5(x^2-36)}{(x-6)(x+6)}=\frac{-5(x-6)(x+6)}{(x-6)(x+6)}$

A point of discontinuity occurs when a number is both a zero of the numerator and denominator.

Since $x=\pm 6$ is a zero for both the numerator and denominator, there is a point of discontinuity there. Since the final function is s(x)=-5 , (6, -5) and (-6, -5) are points of discontinuity.

Report an Error

Example Question #8 : Find A Point Of Discontinuity

Find a point of discontinuity in the following function:

$f(x)=\frac{x^2+x-12}{x^2+2x-8}$

Possible Answers:

There is no point of discontinuity for this function.

(0, 4)

$\left(-2, \frac{5}{7}\right)$

$\left(-4, \frac{7}{6}\right)$

Correct answer:

$\left(-4, \frac{7}{6}\right)$

Explanation:

Start by factoring the numerator and denominator of the function.

$f(x)=\frac{x^2+x-12}{x^2+2x-8}=\frac{(x+4)(x-3)}{(x+4)(x-2)}=\frac{(x-3)}{(x-2)}$

A point of discontinuity occurs when a number is both a zero of the numerator and denominator.

Since x=-4 is a zero for both the numerator and denominator, there is a point of discontinuity there. To find the value, plug in into the final simplified equation.

$\frac{-4-3}{-4-2}=\frac{7}{6}$

$\left(-4, \frac{7}{6}\right)$ is the point of discontinuity.

Report an Error

Example Question #9 : Find A Point Of Discontinuity

Find the point of discontinuity for the following function:

$g(x)=\frac{x^2-1}{x^2+8x+7}$

Possible Answers:

$\left(1, \frac{1}{3}\right)$

There is no point of discontinuity.

$\left(-1, -\frac{1}{3}\right)$

(2, 4)

Correct answer:

$\left(-1, -\frac{1}{3}\right)$

Explanation:

Start by factoring the numerator and denominator of the function.

$g(x)=\frac{x^2-1}{x^2+8x+7}=\frac{(x-1)(x+1)}{(x+1)(x+7)}=\frac{x-1}{x+7}$

A point of discontinuity occurs when a number is both a zero of the numerator and denominator.

Since x=-1 is a zero for both the numerator and denominator, there is a point of discontinuity there. To find the value, plug in into the final simplified equation.

$\frac{-1-1}{-1+7}=-\frac{1}{3}$

$\left(-1, -\frac{1}{3}\right)$ is the point of discontinuity.

Report an Error

Example Question #1 : Find A Point Of Discontinuity

Find the point of discontinuity for the following function:

$h(x)=\frac{x^2-5x+6}{x^2-9}$

Possible Answers:

There is no point of discontinuity for this function.

$\left(6, -\frac{2}{3}\right)$

$\left(5, \frac{1}{8}\right)$

$\left(3, \frac{1}{6}\right)$

Correct answer:

$\left(3, \frac{1}{6}\right)$

Explanation:

Start by factoring the numerator and denominator of the function.

$h(x)=\frac{x^2-5x+6}{x^2-9}=\frac{(x-2)(x-3)}{(x-3)(x+3)}=\frac{x-2}{x+3}$

A point of discontinuity occurs when a number is both a zero of the numerator and denominator.

Since x=3 is a zero for both the numerator and denominator, there is a point of discontinuity there. To find the value, plug in into the final simplified equation.

$\frac{3-2}{3+3}=\frac{1}{6}$

$\left(3, \frac{1}{6}\right)$ is the point of discontinuity.

Report an Error

View Pre-Calculus Tutors

Alexander
Certified Tutor

Champlain College, Bachelor of Science, Game and Interactive Media Design.

View Pre-Calculus Tutors

Garth
Certified Tutor

View Pre-Calculus Tutors

Tom
Certified Tutor

University of Wisconsin-Madison, Bachelor of Science, Atmospheric Sciences and Meteorology. Embry-Riddle Aeronautical Univers...

All Precalculus Resources

12 Diagnostic Tests 380 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

SAT Tutors in Houston, Algebra Tutors in Atlanta, Computer Science Tutors in Washington DC, Reading Tutors in Chicago, GMAT Tutors in San Francisco-Bay Area, SSAT Tutors in Seattle, ISEE Tutors in Phoenix, ISEE Tutors in San Francisco-Bay Area, MCAT Tutors in Denver, Biology Tutors in Phoenix

Popular Courses & Classes

Spanish Courses & Classes in San Francisco-Bay Area, GMAT Courses & Classes in Denver, ACT Courses & Classes in New York City, SSAT Courses & Classes in Dallas Fort Worth, ACT Courses & Classes in Chicago, MCAT Courses & Classes in New York City, MCAT Courses & Classes in Los Angeles, LSAT Courses & Classes in Philadelphia, ISEE Courses & Classes in Miami, SSAT Courses & Classes in Los Angeles

Popular Test Prep

ACT Test Prep in Denver, MCAT Test Prep in Seattle, ACT Test Prep in Chicago, GRE Test Prep in San Francisco-Bay Area, MCAT Test Prep in Houston, ISEE Test Prep in New York City, ACT Test Prep in Miami, GMAT Test Prep in San Diego, GRE Test Prep in Denver, SAT Test Prep in Los Angeles

DMCA Complaint

If you believe that content available by means of the Website (as defined in our Terms of Service) infringes one or more of your copyrights, please notify us by providing a written notice (“Infringement Notice”) containing the information described below to the designated agent listed below. If Varsity Tutors takes action in response to an Infringement Notice, it will make a good faith attempt to contact the party that made such content available by means of the most recent email address, if any, provided by such party to Varsity Tutors.

Your Infringement Notice may be forwarded to the party that made the content available or to third parties such as ChillingEffects.org.

Please be advised that you will be liable for damages (including costs and attorneys’ fees) if you materially misrepresent that a product or activity is infringing your copyrights. Thus, if you are not sure content located on or linked-to by the Website infringes your copyright, you should consider first contacting an attorney.

Please follow these steps to file a notice:

You must include the following:

A physical or electronic signature of the copyright owner or a person authorized to act on their behalf; An identification of the copyright claimed to have been infringed; A description of the nature and exact location of the content that you claim to infringe your copyright, in \ sufficient detail to permit Varsity Tutors to find and positively identify that content; for example we require a link to the specific question (not just the name of the question) that contains the content and a description of which specific portion of the question – an image, a link, the text, etc – your complaint refers to; Your name, address, telephone number and email address; and A statement by you: (a) that you believe in good faith that the use of the content that you claim to infringe your copyright is not authorized by law, or by the copyright owner or such owner’s agent; (b) that all of the information contained in your Infringement Notice is accurate, and (c) under penalty of perjury, that you are either the copyright owner or a person authorized to act on their behalf.

Send your complaint to our designated agent at:

Charles Cohn Varsity Tutors LLC
101 S. Hanley Rd, Suite 300
St. Louis, MO 63105

Or fill out the form below:

Do not fill in this field

Contact Information

* Full legal name

Company name

* Phone number

* Email address

Address

* Address1

Address2

* City

* State

* Zipcode

* Country

Complaint Details

* URL of copyrighted work (where your original material is located)

* Please describe the copyrighted work so that it may be easily identified

I am the owner, or an agent authorized to act on behalf of the owner of an exclusive right that is allegedly infringed.

I have a good faith belief that the use of the material in the manner complained of is not authorized by the copyright owner, its agent, or the law.

This notification is accurate.

I acknowledge that there may be adverse legal consequences for making false or bad faith allegations of copyright infringement by using this process.

* Signature (your digital signature is legally binding)

HIGH SCHOOL

GRADUATE SCHOOL

K-8

Search 50+ Tests

math tutoring

science tutoring

foreign languages

elementary tutoring

other

Search 350+ Subjects

Precalculus : Rational Functions

Study concepts, example questions & explanations for Precalculus

All Precalculus Resources

Example Questions

Example Question #1 : Find A Point Of Discontinuity

Example Question #1 : Find A Point Of Discontinuity

Example Question #1 : Find A Point Of Discontinuity

Example Question #2 : Find A Point Of Discontinuity

Example Question #1 : Find A Point Of Discontinuity

Example Question #6 : Find A Point Of Discontinuity

Example Question #7 : Find A Point Of Discontinuity

Example Question #8 : Find A Point Of Discontinuity

Example Question #9 : Find A Point Of Discontinuity

Example Question #1 : Find A Point Of Discontinuity

All Precalculus Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: